Analytical Study of Anomalous Diffusion by Randomly Modulated Dipole

Abstract: This paper derives the Fokker-Planck (FP) equation for a particle moving in potential by a randomly modulated dipole. The FP equation describes the anomalous diffusion observed in the companion paper [1] and breaks the conservation of the total probability at the singularity by the dipole. It also shows anisotropic diffusion, which is typical in fluid turbulence. We need to modify the probability density by introducing a mechanism to recover the particle. After the modification, the latent fractal dimension matches with the results of the companion paper. We hope that our model gives a new example of fractional diffusion, where the random singularity triggers fractionality.